I’d probably have titled this something including either the term “linear” or “functional analysis”. Because submitted here, we will first interpret “functions” in the context of a function in computer programming, where the statement is more provocative and thus clickbaity.

The problem is many real world functions and problems are nonlinear. But they may have linear components. For example, a dog can be recognized by its outline and the texture of the fur, pattern of the face, etc within that outline. Deep neural nets solve this by composing vector operations, hence the universal approximation theorem and existence of NNs that can recognize dogs (though I could have picked a better example as dog recognition is not continuous I think).

In the context of computer programming, it is not really a helpful statement to say that functions are vectors. But because of the universal approximation theorem and its relatives you could say that “functions are (can be approximated as) compositions of vector operations”